This is just my idea and I cannot claim that it is exactly true. After these assumptions, we do not have any dependency on z (out-of plane direction) coordinate. So, we only end up with a problem on an in-plane surface. Therefore, we should only concentrate on x-y components of stress and strain. What this means, we should not take into account the relations related with the z-coordinate. So, we do not need to take into account the constituve relation of εzz and corresponding stress components.

Thank you for this interesting topic. What I mean with the term relations is in general. Maybe I should even correct my phrase. Under the assumptions that we are making, we should eliminate the terms related with the z-coordinate. As a crude example, like eliminating applied boundary conditions from our global governing equation.So,, we should only have the in-plane related terms in any relation suh as equilibrium equations, constitutive relations,etc. You are right, we should not violate constitutive relation, but here we are making assumptions, so we need to sacrifice something which is OK to ignore under some particular conditions.

The original post made a point. The violation of the 3D constitutive relations is happened because the second assumption is not correct. The first assumption is asymptotically correct for the first approximation of the original 3D model using a 2D model. However, the second assumption, transverse normal remains rigid, is clearly violating the first assumption, plane stress. Because by assuming plane stress, we assume that the plate is free to move in the thickness direction, which means the transverse normal is not rigid. Both assumptions can be valid only if the Poisson's ratio is zero. Recall the well-known and readily observed Poission's effect. The reason the second assumption is used is because it is convient to derive a 2D version kinematics (strain-displacement relations). It is noted that same conflicting assumptions also used to derive beam models dealing with tension and bending: section remain rigid in its own plane and the beam is in uniaxial stress state.

As a final comment, both assumptions are not absolutely needed for one to derive a plate theory. One can use the variational asymptotic method to take advantage of the smallness of theory as the small parameter to reduce the 3D model to a 2D model with the first assumption comes out as a result of the first assumption and the transverse displacement will be a quadratic function of the thickness coordinate. Please refer to the following paper for more details.